试题分析:(1)求
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024088495.png)
的导数,找出
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024104473.png)
处的导数即切线的斜率,由点斜式列出直线的方程即可;(2)求出函数的定义域,在定义域内利用导数与函数增减性的关系,转化为恒成立问题进行求解即可;(3)讨论
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024104491.png)
在定义域上的最值,分情况讨论
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024088495.png)
的增减性,进而解决
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024151650.png)
存在成立的问题即可.
(1)当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023792421.png)
时,函数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024182853.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024197712.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024213831.png)
,曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024104473.png)
处的切线的斜率为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024260676.png)
从而曲线
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在点
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024104473.png)
处的切线方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024307677.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023995524.png)
3分
(2)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240530243381159.png)
令
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024369795.png)
,要使
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在定义域
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024400537.png)
内是增函数,只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024431568.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024400537.png)
内恒成立
由题意
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024463429.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024369795.png)
的图象为开口向上的抛物线,对称轴方程为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024509851.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024525844.png)
, 只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024541560.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024572404.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024587788.png)
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024400537.png)
内为增函数,正实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023870313.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024026485.png)
7分
(3)∵
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023885660.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
上是减函数
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024697358.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024728655.png)
;
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024743323.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024759702.png)
,即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024775715.png)
①当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024790425.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024369795.png)
,其图象为开口向下的抛物线,对称轴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024821473.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024837310.png)
轴的左侧,且
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024853548.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024884332.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
内是减函数
当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024915398.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024931599.png)
,因为
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024884332.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024977563.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025009834.png)
此时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024884332.png)
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
内是减函数
故当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025071432.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
上单调递减
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025118942.png)
,不合题意
②当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025133473.png)
时,由
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025149793.png)
,所以
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240530251651417.png)
又由(Ⅱ)知当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025180369.png)
时,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
上是增函数
∴
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240530252271026.png)
,不合题意 12分
③当
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024572404.png)
时,由(Ⅱ)知
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023823447.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
上是增函数,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025305605.png)
又
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025321442.png)
在
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024681358.png)
上是减函数,故只需
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025367791.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025383491.png)
而
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240530253991432.png)
,
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024728655.png)
即
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/201408240530254301033.png)
,解得
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053025445647.png)
所以实数
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053023870313.png)
的取值范围是
![](http://thumb.1010pic.com/pic2/upload/papers/20140824/20140824053024057871.png)
15分.